Transcript Tutorial # 10 - University of Waterloo

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Tutorial # 10
Morphological Operations
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http://www.youtube.com/watch?v=ICtkU
7I8oZE
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Mathematical morphology
• Based on set theory
▫ In the context of image
processing
 The objects/regions in an
image are the sets
• For binary images
▫ Black region is one set
▫ White region is one set
http://en.wikipedia.org/wiki/File:Venn_A_inte
rsect_B.svg
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Morphological operations
• Local pixel transformations for transforming
images based on region shapes and patterns
• Use logical operations
• Often used on binary images
• Examples:
▫
▫
▫
▫
Erosion
Dilation
Opening
Closing
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Structuring element
• A simple, pre-defined shape
• Used to apply the morphological operations
• Compares the shape with the neighbourhood of
each pixel
• Examples:
Origin
http://www0.cs.ucl.ac.uk/staff/G.Brostow/classes/IP2008/L3_Morphology.pdf
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To use in MATLAB
• Need to specify structuring element – strel()
diamond
disk
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Example from Gonzalez et al. 2008
Set A
Set B
(structuring
element)
X
For display purposes, the
white background is
equivalent to the black of
a binary image while the
purple region is a white
region.
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X
Run B along A so that the origin of B touches each element of A.
At each location, does B completely overlap with B?
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No there is not complete overlap. This pixel in A will not be
included in our final solution.
Let’s move on to its neighbouring pixel.
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X
What happens if B overlaps the entire neighbouring region
of a pixel in A?
This pixel becomes part of the new solution.
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We can continue like this for every single pixel.
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The resulting solution gives us something like this. It’s a
smaller version of the original set A. This is referred to as
erosion.
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http://www.youtube.com/watch?v=fmyE7
DiaIYQ
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Erosion
• Morphological operation that erodes or shrinks
the boundaries of regions of some foreground
pixels (ie: white pixels)
• This equation basically states:
The erosion of A by B are the points z such that
the translation of B by z is contained in A
(Gonzalez et al. 2008)
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Erosion Example
Original image
Binary image
http://www.bristol-business.net/wpcontent/uploads/2012/01/microchip.jpg
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Erosion Example
Original binary image
Eroded image with
structuring element (square
3x3)
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Erosion Example
Original binary image
Eroded image with
structuring element (square
5x5)
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Uses
• Can be used to filter out image details smaller
than the structuring element (as seen in previous
example
• Reduce thickness of objects
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Dilation
• Opposite of erosion
• Causes expansion of foreground boundaries
• This time, the structuring element B is reflected
about its origin and is shifted by z
• The dilation is then all the elements of A which
overlap with by at least one element.
• Since B is often symmetrical, and B are equal.
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X
Using the same example as before, let’s see what happens
when B is moved along A. If any part of B is touching A,
that pixel is added to the final solution.
In this case, the pixel is not added, since there is no
overlap with A.
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X
On this second pixel, there is overlap with one pixel. Then
we add the pixel in A at the origin.
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The dilation of A by B is now
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http://www.youtube.com/watch?v=xO3E
D27rMHs
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Dilation Example
Original binary image
Eroded image with
structuring element (square
3x3)
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What if we used a different structuring
element?
Original binary image
Eroded image with which
structuring element?
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What if we used a different structuring
element?
Original binary image
Eroded image with
structuring element (disk,
r =10)
Note the rounded edges because
a disk was used.
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Uses
• Can be used to bridge gaps between objects
• Instead of blurring an image using a lowpass
filter, dilation results in a binary image
Image with weak or broken edges
Dilation connects the gaps
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Zoomed in
Image with weak or broken edges
Dilation connects the gaps (would
require a large structuring element for
complete connection)
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How can I use this in MATLAB?
• Erosion – imerode()
• Dilation – imdilate()
• General (includes many of the different
morphological operations) – bwmorph()
• Structuring element – strel()
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Combinations
• Erosion and dilation can then be combined to
create two other morphological operations:
▫ Opening
▫ Closing
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Opening
• Smoothes object contours
• Removes thin protrusions
• Erosion followed by dilation
• Opening can be thought of as the
structuring element B, as a “rolling
ball”
• The new boundary is found when B is
“rolled” inside the boundary
• (Dark blue is the original, Red circles
are the structuring elements and light
blue is the resulting opening of A by B
http://en.wikipedia.org/wiki/Fi
le:Opening.png
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Advantages over erosion
Original noisy image
Eroded image
(disk, r = 1)
Opening of image
(disk, r = 1)
Erosion can remove the undesired noise however also affects all of the desired
foreground pixels. Opening retains the foreground pixels which the structuring
element can roll around in.
Image courtesy of MATLAB
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Closing
•
•
•
•
•
Smoothes object contour
Fuses narrow breaks
Eliminates small holes
Fills gaps in the contour
Dilation followed by erosion
• Closing can be thought of as rolling B on
the outside of A
• The new boundary is found when B is
“rolled” outside the boundary
• (Dark blue is the original, Red circles
are the structuring elements and all blue
regions are the resulting closing of A by
B
http://en.wikipedia.org/wiki/Fi
le:Closing.png
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Closing vs Dilation
Original image
(want to remove
small holes)
Dilated image
(disk, r = 10)
Closing of image
(disk, r = 10)
http://homepages.inf.ed.ac.uk/rbf/HIPR2/close
.htm
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Example: Segmentation
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Example
Binary
Opened
Dilated
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Example: Segmentation
Original image
Closing the thresholded image
http://homepages.inf.ed.ac.uk/rbf/HIPR2
/close.htm
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Example: Border Irregularity
Idea: Capture “spiky” border irregularity
𝑓1𝐵
𝐴𝑐𝑙𝑜𝑠𝑒𝑑 − 𝐴𝑙𝑒𝑠𝑖𝑜𝑛 𝐴𝑙𝑒𝑠𝑖𝑜𝑛 − 𝐴𝑜𝑝𝑒𝑛
=
+
𝐴𝑙𝑒𝑠𝑖𝑜𝑛
𝐴𝑙𝑒𝑠𝑖𝑜𝑛